Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach
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- Math. Comp. 69 (2000), 103-120 Request permission
Abstract:
In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the $H^1, L^2$ norms and new results in the max-norm. For the elliptic problems we demonstrate that the error $u-u_h$ between the exact solution $u$ and the approximate solution $u_h$ in the maximum norm is $O(h^2|\ln h|)$ in the linear element case. Furthermore, the maximum norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.References
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Additional Information
- So-Hsiang Chou
- Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221, U.S.A.
- Email: chou@zeus.bgsu.edu
- Qian Li
- Affiliation: Department of Mathematics, Shandong Normal University, Shandong, China
- Received by editor(s): March 19, 1996
- Received by editor(s) in revised form: April 22, 1996
- Published electronically: August 25, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 103-120
- MSC (1991): Primary 65F10, 65N20, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-99-01192-8
- MathSciNet review: 1680859